Download ON DERIVING LUMPED MODELS FOR BLOOD FLOW

MATHEMATICAL BIOSCIENCES
AND ENGINEERING
Volume 1, Number 1, June 2004
http://math.asu.edu/˜mbe/
pp. 61–80
ON DERIVING LUMPED MODELS FOR BLOOD FLOW AND
PRESSURE IN THE SYSTEMIC ARTERIES
Mette S. Olufsen
Department of Mathematics
North Carolina State University
Box 8205
Raleigh, NC 27695
Edwardsville, Illinois, 62026-1653
Ali Nadim
Keck Graduate Institute
535 Watson Drive
Claremont, CA 91711
(Communicated by James F. Selgrade)
Abstract. Windkessel and similar lumped models are often used to represent blood flow and pressure in systemic arteries. The windkessel model was
originally developed by Stephen Hales (1733) and Otto Frank (1899) who used
it to describe blood flow in the heart. In this paper we start with the onedimensional axisymmetric Navier-Stokes equations for time-dependent blood
flow in a rigid vessel to derive lumped models relating flow and pressure. This
is done through Laplace transform and its inversion via residue theory. Upon
keeping contributions from one, two, or more residues, we derive lumped models of successively higher order. We focus on zeroth, first and second order
models and relate them to electrical circuit analogs, in which current is equivalent to flow and voltage to pressure. By incorporating effects of compliance
through addition of capacitors, windkessel and related lumped models are obtained. Our results show that given the radius of a blood vessel, it is possible
to determine the order of the model that would be appropriate for analyzing the flow and pressure in that vessel. For instance, in small rigid vessels
(R < 0.2 cm) it is adequate to use Poiseuille’s law to express the relation
between flow and pressure, whereas for large vessels it might be necessary to
incorporate spatial dependence by using a one-dimensional model accounting
for axial variations.
1. Introduction and background. Windkessel and similar lumped models are
often used to represent blood flow and pressure in the arterial system [4, 26, 29, 34].
These lumped models can be derived from electrical circuit analogies where current
represents arterial blood flow and voltage represents arterial pressure. Resistances
represent arterial and peripheral resistance that occur as a result of viscous dissipation inside the vessels, capacitors represent volume compliance of the vessels
that allows them to store large amounts of blood, and inductors represent inertia
of the blood. The windkessel model was originally put forward by Stephen Hales
in 1733 [13] and further developed by Otto Frank in 1899 [11]. Frank used the
2000 Mathematics Subject Classification. 76205, 92C35.
Key words and phrases. Arterial modeling, Lumped arterial models.
61
62
METTE S. OLUFSEN AND ALI NADIM
windkessel model to describe blood flow in the heart and systemic arteries. He
used the analogy of an old-fashioned hand-pumped fire engine (in German “windkessel” pump). Firemen pump water into a high-pressure air-chamber by periodic
injections at high pressure. When the air-chamber is full, the high pressure drives
the water out in a steady jet. This analogy starts at the left ventricle where the
blood pressure varies from a low of nearly zero to a high of approximately 120
mmHg and continues into the aorta and the systemic arteries where the pressure
variation is significantly less because of the elasticity of the large systemic arteries.
This analogy resulted in the development of the original (two-element) windkessel
model comprising an electrical circuit with one resistor and one capacitor.
Even though the model was originally derived for the ventricle and the aorta,
it was also used to describe blood flow in the systemic arteries alone (i.e. without
explicitly including the heart); in this case the capacitor represents the compliance
of the large arteries while the resistor represents the resistance of the small arteries
and arterioles (so-called resistance vessels). The two-element windkessel model was
later extended to the three-element windkessel model, which has two resistors and a
capacitor. In this model, the additional resistor is thought to represent the characteristic impedance of the aorta and the large compliance vessels. The three-element
windkessel model is widely used, and it produces realistic blood flow and pressure
wave shapes as well as estimates experimental data [7, 10, 16, 21, 28, 30, 35]. Further expansions of the windkessel model into a four-element model have proven even
better at getting good comparisons between measured blood flow and pressure. In
addition to the two resistors and the capacitor of the three-element windkessel
model, the four-element model includes an inductor representing the inertia of the
blood [16, 29, 31]. The advantage of lumped models is that they are easy to understand and solve, since they give rise to simple ordinary differential equations.
However, these models include a number of parameters (resistors, capacitors, and
inductors), and it is not obvious how to estimate the parameters from measurements
of arterial blood flow and pressure [21, 25, 29].
In general, blood flow in arteries is a pulsatile flow in tapered elastic vessels that
are connected in a branching network [5, 14, 19, 20, 24, 27]. Blood flow is unsteady,
and the fluid is non-Newtonian. So, to develop a complete model for blood flow
in arteries, these effects should be taken into account. That is, one would have
to use the full theory of fluid dynamics, which requires solving the Navier-Stokes
(NS) equations, together with appropriate non-Newtonian constitutive relations
for blood, coupled with the dynamics of the compliant vessels through which blood
flows. For the large arteries, blood can usually be modelled as incompressible and
Newtonian; nevertheless, to study blood flow in detail, it is still necessary to solve
the NS equations.
The disadvantage of working with the full non-linear NS equations is that, even
if only one spatial dimension (axial) is taken into account, it is significantly more
difficult to set up a system of equations predicting the blood flow and pressure in
all of the large systemic arteries [19, 20]. In addition, while such one-dimensional
models provide insight into wave-propagation and some of the non-linear dynamics,
they are not useful for answering questions aimed at understanding the global
behavior of the system. One-dimensional models can be obtained by assuming that
the vessels are axisymmetric and that the velocity profile across the vessel diameter
is known [5, 14, 19, 20, 27].
ON LUMPED MODELS FOR BLOOD FLOW
63
There are many applications, however, where it would be important to be able
to solve the model equations for blood flow in arteries in real time; e.g., in the development of anesthesia simulators [17], where a mathematical model of the cardiovascular system is used in conjunction with various pharmaco-kinetic and dynamic
models. In such cases, use of lumped models provides a distinct advantage. Other
reasons for using lumped models instead of the NS equations to analyze data that
include dynamic changes, such as those associated with posture change from sitting
to standing [21], or baroreceptor regulation [9, 22, 23, 32, 33]; finally such models
are useful because they are easy to implement in a clinical setting.
Womersley was the first person to study blood flow in arteries using a linearized
version of the NS equations [36, 37]. This system of equations is discussed further
by Atabek, Lew, and Gessner in [2, 3, 12]. The approach used in this paper is based
on the ideas outlined by Gessner [12] and Berger [5] with the aim of comparing the
windkessel model with the fluid dynamic equations.
In the present work we use the equations of fluid dynamics to derive a number of
lumped models for blood flow and pressure in the systemic arteries. More precisely,
starting with the one-dimensional NS equations describing time-dependent flow
and pressure in a rigid vessel, we derive first- and second-order lumped models
using Laplace transform, residue theory, and solutions to a Bessel equation. In
the inversion step of the Laplace transform, by including the residues from more
and more poles, we can obtain successively higher-order models. The resulting
lumped models can be represented by electrical circuits. It should be noted that the
immediate interpretation of these lumped circuits has to be for the particular vessel
for which the one-dimensional model was derived. We show that the differential
equations representing the windkessel model have the same form as those obtained
from a first-order approximation of the fluid dynamics equations for flow in a rigid
vessel, but that the parameters must be interpreted somewhat differently. The
novelty in our approach compared to earlier work is that by using the Laplace
transform and residue theory, we obtain the solution exactly in the Laplace domain
and we make systematic approximations only during the inversion step to derive
various lumped models that are appropriate to the diameter of the blood vessel
being studied. Based upon these results, we can assess when it would be appropriate
to treat a vessel as a lumped system and when it would be better to use a distributed
(e.g., one-dimensional) model.
Our results suggest the following based on physical properties of blood: For
vessels with a radius smaller than approximately 0.2 cm, effects of inertia can
be neglected, and if the vessels are rigid it is adequate to use Poiseuille’s simple
relation between pressure drop and flow rate. For rigid vessels with a radius in the
approximate range between 0.2 and 0.5 cm, either a first- or a second-order lumped
model can be used to relate flow and pressure. For rigid vessels with a radius in the
approximate range between 0.5 and 1.5 cm, higher-order terms should be included.
A reduced second-order model would be appropriate, depending on the magnitude
of the time-scales that appear within third- or higher-order models. We do not
derive these higher-order models explicitly, but they can be found easily using the
approach outlined in this paper. For non-rigid vessels, capacitors can be added to
account for the vessel compliance. Finally, for vessels whose radii are larger than
approximately 1.5 cm, it would be more appropriate to take spatial dependence into
account and to model the relation between pressure and flow using a distributed
(e.g., a one-dimensional) model [14, 19, 20, 24, 27].
64
METTE S. OLUFSEN AND ALI NADIM
r
p(0, t)
R
p(L, t)
x
L
0
Figure 1. The vessel, the radius of the vessel is R and the length
is L. The pressure at the inlet into the vessel is p(0, t) and the
pressure at the outlet is p(L, t) = p(0, t) − (∆p)p∗ (t).
In the remainder of this paper we will first discuss (in Section 2) how the NavierStokes equations for an incompressible and Newtonian fluid through a rigid vessel
can be approximated by ordinary differential equations. In Section 3 we will describe how these ordinary differential equations can be interpreted as lumped models
that can be represented by electrical circuit analogies. This section has three parts;
the first part shows how our model can be represented by corresponding lumped
models, the second part compares our model with the windkessel models, and the
last part discusses how elasticity, represented by capacitors, can be included into
the lumped models. Finally, in Section 4 we will discuss our results.
2. Derivation of a lumped mathematical model for blood flow in arteries.
Blood flow in arteries can be modelled as a one-dimensional axisymmetric flow of
an incompressible and Newtonian fluid through a rigid vessel, for which the NS
equations simplify to:
µ ∂
∂u
∂u ∂p
+
=
r
,
(1)
ρ
∂t
∂x
r ∂r
∂r
where u(r, t) is the longitudinal velocity, ρ = 1.06 g/cm3 is the density, µ = 0.049
g/cm/s is the viscosity, and ν = µ/ρ = 0.046 cm2 /s is the kinematic viscosity of
the blood. The pressure p(x, t) inside the vessel is assumed to be constant over
the cross-sectional area (independent of the radial coordinate r). Equation (1)
describes time-dependent flow and pressure in a rigid vessel and is often referred
to as Womersley’s equation [36, 37]. It is obtained from the NS equations by
assuming that the flow is unidirectional and axisymmetric. The assumption of
unidirectional flow applies as long as the vessel is straight and sufficiently long to
ignore flow disturbances at the inlet and the outlet of the vessel. Nonlinear inertial
effects are absent provided that the flow remains laminar, requiring the Reynolds
number to be below a critical value [8, 18, 24]. Even when the vessel is compliant,
for long wavelength variations of flow and pressure, the same simplified equation
approximately applies.
Our assumption of a velocity u(r, t) that is independent of x gives that pressure is
varies linearly with distance along the vessel (see Fig. 1). Hence, the characteristic
pressure gradient can be written as
−
p(0, t) − p(L, t)
∆p ∗
∂p
=
=
p ,
∂x
L
L
(2)
ON LUMPED MODELS FOR BLOOD FLOW
65
where p∗ is the non-dimensional pressure gradient, ∆p is the characteristic change in
pressure, and L is the length of the vessel. Equation (1) can be non-dimensionalized
using the following dimensionless variables:
tν
r
u(r, t)Lµ
t∗ = 2 ,
r∗ = ,
and
u∗ (r∗ , t∗ ) =
.
(3)
R
R
∆pR2
Inserting equations (2) and (3) into equation (1) and reusing the original symbols
yield
∂u 1 ∂u ∂ 2 u
−
− 2 = p(t).
∂t
r ∂r
∂r
This equation can be transformed into a Bessel equation that can be solved analytically using the Laplace transform
Z ∞
L{u(r, t)} =
u(r, t) e−st dt = u(r, s)
0
and the property
L
The Laplace transform gives
∂u
∂t
= su(r, s) − u(r, 0).
√
d2 u 1 du
+
+ (i s)2 u = −p(s),
(4)
2
dr
r dr
√
where i = −1, p(s) = L(p(t)), and it is assumed that there is no flow initially
(i.e. u(r, 0) = 0). The particular solution is
p(s)
.
s
The remaining homogeneous equation is equivalent to Bessel’s equation:
u(r, s) =
d2 u
du
+y
+ y 2 u = 0.
2
dy
dy
Hence, the general solution to the differential equation (4) can be written as
√
√
p(s)
,
u(r, s) = c1 J0 (ir s) + c2 Y0 (ir s) +
s
where c1 and c2 are arbitrary constant and J0 and Y0 are the zeroth-order Bessel
functions. Since Y0 is singular at r = 0, constant c2 must be set to zero. Applying the no-slip boundary condition u(1, s) = 0 makes it possible to solve for the
remaining constant c1 to yield
√ J0 (ir s)
p(s)
√
u(r, s) =
1−
.
(5)
s
J0 (i s)
y2
The volumetric flow rate (or simply the flow) is defined in dimensionless form by
Z 1
q(s) = 2π
u(r, s) r dr.
(6)
0
Inserting the velocity in equation (5) into equation (6) gives
√
πp(s)
2J1 (i s)
√
√
q(s) =
1−
,
s
i sJ0 (i s)
which can be written as
√
π
2J1 (i s)
√
q(s) = p(s)K(s),
K(s) =
1− √
.
s
i sJ0 (i s)
(7)
(8)
66
METTE S. OLUFSEN AND ALI NADIM
Computing q(t) using equation (7) involves computing the inverse Laplace transform of q(s), and hence, inversion of K(s). This inversion can be done using the
method of residues, which requires finding the poles of K(s). The inverse transform is found by closing the inversion contour integral by a semi-circle in the left
half-plane and letting its radius tend to infinity, where the contribution from the
semi-circle itself vanishes. To complete the inversion it is advantageous to write
K(s)
K(s)
[sp(s)] = M (s) [sp(s)] ,
M (s) =
.
s
s
In the time domain the flow q(t) can then be found by applying the convolution
theorem
Z t
q(t) =
M (t − t̃)p0 (t̃) dt̃
(9)
q(s) =
0
since
p0 (t) = L−1 {sp(s)},
if p(0) = 0. The kernel M (t) is given by
M (t)
L−1
=
1
2πi
X
=
=
K(s)
s
c+i∞
K(s) st
e ds
s
c−i∞
K(s) st
e
.
Res
s
Z
Upon finding the residues (see Appendix A), the kernel in the time domain is found
to be given by the infinite series
M (t) =
2
∞
X
e−βn t
π
− 4π
,
8
βn4
n=1
(10)
where the βn ’s are the roots of the Bessel function J0 (βn ) = 0. The first few of these
roots are given by β1 = 2.40483, β2 = 5.52008, and β3 = 8.65373 [1]. Inserting
equation (10) into the convolution integral in equation (9) gives
Z t
∞
X
2
1
π
q(t) = p(t) − 4π
e−βn (t−t̃) p0 (t̃) dt̃.
(11)
4
8
β
n=1 n 0
This equation is one of the main results of the analysis; in the following we seek to
approximate this solution to obtain a simple differential equation relating flow and
pressure. A convenient way to proceed is to define
Z t
2
fn (t) =
e−βn (t−t̃) p0 (t̃) dt̃
(12)
0
so that upon differentiation
fn0 (t) + βn2 fn (t) = p0 (t).
(13)
Inserting equation (12) into equation (11) gives
q(t) =
∞
X
π
fn (t)
p(t) − 4π
8
βn4
n=1
(14)
ON LUMPED MODELS FOR BLOOD FLOW
67
with each fn (t) obtained by solving equation (13). Truncating the series in equation
(14) at n = 1 gives
π
4πf1 (t)
q(t) = p(t) −
.
(15)
8
β14
To solve equation (15), it is necessary to get an expression for f1 (t). Such an
expression can be found by introducing the operator D = d/dt and using it to solve
equation (13) for n = 1:
f1 (t) = (D + β12 )−1 Dp(t).
Inserting equation (16) into equation (15) and applying the operator (D +
to both sides of the equation gives the differential equation
1 dq
π A dp
32
+q =
+ p , where A = 1 − 4 .
β12 dt
8 β12 dt
β1
(16)
β 12 )/β12
(17)
Equation (17) can be written in the form
dq
π
λq
+q =
dt
8
where
dp
λp
+p ,
dt
(18)
A
1
≈ 0.1729 and λp = 2 = Aλq ≈ 0.0075.
2
β1
β1
Equation (18) has a form similar to Jeffrey’s model that describes linear viscoelasticity [6] and by analogy the constant λq can be interpreted as the relaxation time
and λp as the retardation time. Note that λp λq , indicating that the change in
pressure with time has a much smaller effect than the pressure itself. This equation
represents the first-order model; at the end of this section, we interpret it as a
lumped parameter model.
This method can also be used to obtain a second-order model by including two
terms of the sum in equation (14). For the second-order method two equations
of the form of equation (13) would have to be solved, one for f1 (t) and one for
f2 (t). Similar to the first-order model, equations for these two expressions can be
found using the operator method. Inserting the results into equation (14), and
applying the operator (D + β12 )(D + β22 )/β12 β22 to both sides of the equation yields
the differential equation
2
1 d2 q
1
1 dq
π
1
1
1
d p
+
+ 2
+q =
1 − 32
+ 4
+
β12 β22 dt2
β12
β2 dt
8 β12 β22
β14
β2
dt2
1
dp
1
1
1
+ 2 − 32
+ 6
+p .
(19)
β12
β2
β16
β2
dt
λq =
The time-scales for the second-order model can be found by solving the characteristic polynomials for the second-order operators on each side of equation (19). In
other words, writing
q(t)
=
cq1 e−t/λq1 + cq2 e−t/λq2
p(t)
=
cp1 e−t/λp1 + cp2 e−t/λp2
as the solutions of the homogeneous differential equations obtained by setting each
side of equation (19) to zero yields quadratic equations for (1/λqi ) and (1/λpi ) for
i = 1, 2. These time-scales are similar to the retardation time and the relaxation
time, except that we have two of each. Consequently, in the following we will refer
to the “pressure” time-scales and the “flow” time-scales. The equation for the flow
68
METTE S. OLUFSEN AND ALI NADIM
0.25
τ
λ =λ
q q1
λ
p
λ
q2
λ
p1
λ
0.2
0.15
τ
p2
0.1
0.05
0
0
0.5
R [cm]
1
1.5
Figure 2. The dimensionless cardiac cycle τ (blue) as a function
of the vessel radius R. The horizontal lines show the flow timescales λq ,λq1 , and λq2 (red) and the pressure time-scales λp , λp1 ,λp2
(green).
time-scales is easy to solve analytically, but the equation for the pressure timescales leads to long expressions. As a result the pressure time-scales are computed
numerically. The four time-scales are:
1
1
(20)
λq1 = 2 = 0.1729 λq2 = 2 = 0.0328
β1
β2
λp2 = 0.0013.
λp1 = 0.0378
An interesting observation based on examining the above results is that the secondorder model for the flow includes the same time-scale λq1 as the first-order model
plus an additional time-scale λq2 . The same is not the case for the pressure timescales since both time-scales are different from the first-order model. Note that
the biggest pressure time-scale for the second-order model is larger than the one
obtained from the first-order model.
An important insight that can be gleaned from such derivations is related to the
time-scales associated with the first (18), second (19), and higher-order equations.
One can study these in relation to the characteristic time-scale of the cardiac cycle.
Let the heart-rate be defined by HR so that the period of the cardiac cycle is
T = 1/HR= 1/f where f is the frequency. Then, the angular frequency is given
by ω = 2πf giving a time-scale 1/ω = T /(2π), which can be non-dimensionalized
using the time-scale for t∗ (see equation (3)). Thus, the dimensionless period of a
cardiac cycle can be written as
Tν
τ=
.
2πR2
Fig. 2 shows τ as a function of the vessel radius for a heart-rate of 60 beats/min (or
1 beat/sec) as well as the six different time-scales calculated thus far. The figure
shows that for vessels with a radius smaller than approximately 0.2 cm, τ (blue)
is larger than all flow time-scales λq , λq1 , λq2 (red) and all pressure time-scales
λp , λp1 , λp2 (green). As a result, a time-variation of pressure on the time-scale
ON LUMPED MODELS FOR BLOOD FLOW
69
τ causes a change in flow that would occur almost instantaneously, the changes
resulting from inertia occurring on a time-scale shorter than one cardiac cycle and
therefore being almost negligible. The flow will be quasi-steady, and effects of inertia
can be neglected. If the vessels are rigid, they will act as pure resistance vessels,
where p is proportional to q; i.e., effects of inertia can be neglected. (Again, it
should be noted that our models do not include compliance. However, if compliance
is included, as discussed in Section II.C., a similar analysis of time-scales can be
performed.) For these small vessels, the time derivatives in equations (18) or (19)
are negligible, simplifying the equations to
π
q ≈ p.
8
so that we do not see any inertial effects. In dimensional form one recovers
Poiseuille’s relation between flow and pressure for steady laminar flow in a rigid
vessel (as expected):
πR4 p
(21)
q≈
8µL
using
µL
p
q∗ =
q and p∗ =
.
(22)
∆pR4
∆p
Note that the pressure p(t) actually represents the pressure difference between the
inlet and the outlet of the vessel (see equation (2) and Fig. 1).
Rigid vessels with a radius R > 0.2 cm the flow can no longer be treated as
quasi-steady and it is necessary to include time-derivatives. For vessels with a
radius 0.2 < R < 0.5 cm, the time-scales that appear only in the second-order
model λq2 (red dotted line), λp1 , and λp2 (green dotted lines) can be neglected, and
we can approximate the flow in the vessel using the first-order model. However,
since λq1 > τ (red solid line), terms associated with this time-scale should be
included, but the time-scale dependent on λp (green solid line) can be neglected.
Consequently, it should be sufficient to model flow in such a rigid vessel using the
first-order differential equation
π
dq
+ q = p.
(23)
dt
8
For vessels with a radius R > 0.5 cm, our analysis suggests that a secondorder model would be more appropriate including all derivatives for both flow and
pressure. Rewriting equation (19) in terms of the flow time-scales gives:
dq
dp
π
d2 q
d2 p
λq1 λq2 2 + (λq1 + λq2 ) + q =
λp1 λp2 2 + (λp1 + λp2 ) + p . (24)
dt
dt
8
dt
dt
λq
However, since λp2 λp1 and since λp2 is very small, the term involving the
second-order derivative in pressure will be significantly smaller than τ for vessels
with a radius 0.5 < R < 1.5 cm and may thus be neglected. (This, of course,
requires the time-derivatives of pressure not to be so large as to compensate for
the small coefficients in front of those derivatives. The implicit assumption is that
these variations occur in a time-scale of order τ .) Finally, for the very large vessels,
such as the aorta, where the radius may exceed 1.5 cm, it is necessary to include all
terms for both flow and pressure time-scales. In fact, for the large vessels an even
higher-order model is called for. Solely based on the time-scales λq , a sixth-order
model would be needed to obtain a λq < τ for R = 1.5 cm.
70
METTE S. OLUFSEN AND ALI NADIM
In analyzing the time-scales as was undertaken above, it is worthwhile to keep
the following general structure in mind. The relation between pressure and flow
in the time domain ordinarily takes the form D1 q(t) = D2 p(t) where D1 and D2
are linear differential operators (of successively higher order, as needed). First,
one has to decide which of the two variables q(t) and p(t) should be regarded as
input, and which as output. For instance, if p(t) is taken to be the input, the righthand side of the differential equation will be known, regardless of the time-scales
appearing therein, and the equation is to be solved for q(t). Normally, we regard
p(t) and q(t) as periodic functions of time so that issues related to imposing an
appropriate number of initial conditions (depending on the order of the differential
operator) do not come into play. What does matter, however, is the degree to
which the differential operator on each side could be reduced. The analysis of timescales addresses this issue. For instance, if the pressure forcing is characterized by
radian frequency ω, say, so that one should expect the flow response to also be
characterized by the same frequency, then in each factor of the form (1 + λd/dt)
in each differential operator, one can decide whether to keep or omit the derivative
term in comparison to unity. When λω 1, those factors in the overall differential
operators D1 and D2 may be replaced by unity, reducing the order of the equation.
As such, by considering successively higher-order differential models, factoring the
operators as products of individual terms of the form (1+λj d/dt) (to within a single
multiplicative constant outside these factors), and by omitting those factors for
which λj ω is small in comparison to unity, one can systematically decide what the
order of the appropriate model should be. An issue that complicates this analysis,
however, is that for each higher-order model, the time-scales turn out to be distinct
from the ones found at the previous lower order. So without actually obtaining
the higher-order model, it would be hard to asses whether the lower order model is
adequate. One way to avoid this complication is to use the pair of equations (13)
and (14) as the starting point. Since for each n, equation (13) involves a single
operator of the form (1 − λn d/dt) with λn = 1/βn2 , one can determine, based on the
time-scale in p0 (t), how many of these equations should be regarded as differential
equations so that the rest can be regarded as algebraic equations directly relating
fn (t) to p0 (t).
3. Lumped models for arterial blood flow.
3.1. Derivation of lumped arterial models. As discussed in the introduction,
a number of lumped models have been used to describe flow and pressure in the
systemic arteries. The most popular of these models is the three-element windkessel
model with two resistors and a capacitor (see Fig. 3D). More recently it has been
suggested that addition of an inductor to the windkessel model, representing the
inertia of blood, improves the agreement between the model and actual data [29].
Our derivation of the differential equation relating flow and pressure was based
on modeling the NS equations for flow in a rigid vessel. As a result it is not possible
to obtain lumped models that include a capacitor, but only models that include
inductors and resistors representing the resistance to the flow and the inertance of
the blood. Also, it should be noted, that our derivation above should be interpreted
as a lumped model representing the rigid vessel it was derived for. Similar models
could be obtained for any of the systemic arteries, and they could be added in
series to provide a system of lumped models representing the arterial tree. Such
systems have been derived and are often referred to as transmission line models.
ON LUMPED MODELS FOR BLOOD FLOW
71
The systems models can still be viewed from the same point of departure, since
they essentially think of the entire system as flow running through one vessel with
appropriate dimensions.
From the analysis in the previous section (see Fig. 2) one observes that for rigid
vessels with a radius smaller than 0.2 cm, effects of inertia can be ignored and the
vessels can be modelled as resistance vessels. In people, arteries of that caliber
are almost rigid, so it would not be necessary to add a capacitor, only a resistor.
However, if a model is constructed to study the rat aorta it is necessary to include
a capacitor representing the compliance of this vessel. The circuit corresponding to
these vessels is shown in Fig. 3A. The corresponding equation for this circuit is:
p(t) = R1 q(t).
Comparing this equation with equation (21) gives R1 = 8µL/πR4 . For vessels with
a radius between 0.2 < R < 0.5 cm. The flow in the vessel can be modelled using
equation (23). Using the scaling factors given earlier (in equations (3) and (22))
equation (23) can be re-dimensionalized as follows
πR4
λq R2 dq
+q =
p,
(25)
ν dt
8µL
where again p(t) is the pressure difference between the inlet and the outlet of the
vessel, so ∂p/∂x = −p/L. The above equation can be obtained from a circuit with
a resistor and an inductor in series (see Fig. 3B). For this circuit the equation is:
L1 dq
1
+q =
p.
(26)
R1 dt
R1
Comparing the last equation with equation (25) gives values for the inductance L 1
and the resistance R1 as functions of the vessel and fluid properties:
8µL
8λq ρL
R1 =
and L1 =
.
(27)
4
πR
πR2
This result is similar to the one obtained by Stergiopulos et al. [29]. The difference
is that they have a factor 4/3 instead of 8λq , which is slightly higher. Including the
retardation term (even though it is small) yields a differential equation in the form
of equation (18), which in dimensional form can be written as
πR4 λp R2 dp
λq R2 dq
+q =
+p .
(28)
ν dt
8µL
ν dt
This differential equation can be represented by two resistors and an inductor (see
Fig. 3C). For this circuit the equation is:
1
1 dq
1
L1 dp
+
+q =
+p .
(29)
L1
R1
R2 dt
R2 R1 dt
Comparing this with equation (28) gives
8ρL
(λq − λp ) 2
πR
8µL
λq
−1
(30)
R1 =
λp
πR4
8µL
R2 =
.
πR4
Note that our derivations of the differential equations (25) and (28) were based on
including only the first term from the sum on the right hand-side of equation (14).
L1
=
72
METTE S. OLUFSEN AND ALI NADIM
A:
R1
p1 = 0
p
q
B:
R1
p1
p
L1
p2 = 0
q
C:
q̃
p
R1
p1
R2
p2 = 0
q
q
L1
q − q̃
q̃
D:
R1
p1
p
R2
p2 = 0
q
q − q̃
C1
E:
q̃
p
R1
L1
q̃˜
p1
C1
R2
p2 = 0
q − q̃˜
q − q̃
Figure 3. Lumped models: A: A resistor and an inductor. B:
Two resistors and an inductor. C: Three-element windkessel model
with two resistors and a capacitor. D: Four-element windkessel
model with two resistors, an inductor, and a capacitor.
For vessels with a radius R > 0.5 cm, as discussed earlier, it is necessary to use
the second-order model in equations (19) or (24). Upon examining this equation
ON LUMPED MODELS FOR BLOOD FLOW
73
it becomes clear that the coefficients in front of both second-order derivatives are
small and can thus easily be neglected. In addition, the first-order derivative of
p has a small coefficient. So, neglecting these terms yields a differential equation
in the same form as equation (23). The difference between the equation obtained
from the second-order model and equation (23) is that the time-scale for λ q will
be changed from λq = 1/β12 to λq = λq1 + λq2 = (1/β12 + 1/β22 ), increasing λq
from 0.1729 to 0.2057. Consequently, the parameter for L1 in equation (27) will be
increased, whereas the resistance parameter R1 will remain unchanged. Including
the first-order derivative in p (the largest of the terms that were neglected) will
yield an equation of the form of equation (18), where λq = λq1 + λq2 = 0.2057 and
λp = λp1 + λp2 = 0.0392. The analogous circuit is the one shown in Fig. 3C with
equation (29). The parameters in equations (30) would be modified such that the
inductor L1 would be slightly bigger, the resistance R1 significantly smaller, but the
resistance R2 would remain unchanged. Including the second-order derivative in q
would amount to inserting a second inductor in series with the second resistance R 2
in Fig. 3C. Reflecting on our analysis above of the second-order model, it is our belief
that for vessels with a radius smaller than 1.5 cm it is not necessary to include terms
of higher than second order even though the time-scale for p may increase, since
the coefficients of high-order derivatives in the differential equation will be small.
For vessels with a radius larger than 1.5 cm, we believe, as stated in the previous
section, that a higher-order model becomes necessary. In fact we would recommend
modeling such large vessels with a one-dimensional fluid dynamics model.
It should be noted that the two differential equations models discussed above are
both based on flow in a rigid vessel. The arteries are compliant; so, as discussed
in the introduction to this section, elasticity has to be included in our model. Including elasticity is not trivial if the goal is to derive lumped models directly from
the fluid dynamic equations. In the work by Bergel in [5], a continuity equation
is added, however as discussed in section II.C these considerations do not lead to
ordinary differential equations that can be represented by circuits including capacitors. In other words, it is possible to derive equations with compliance, but not to
obtain the very popular windkessel models. One way to account for the effect of
elasticity effects is by superposition of the results for the rigid vessel with a compliant component represented by a capacitor in either of the models in Fig. 3B or C
(or even A, if the small resistive vessels are also known to be compliant such as the
rat aorta). In fact, the most promising lumped model for blood flow and pressure
in the systemic arteries is the four-element windkessel model (see Fig. 3E) obtained
by adding a capacitor to the circuit in Fig. 3C.
3.2. Windkessel models. In the following we relate the differential equations we
have obtained to the popular windkessel model, and in the next section we provide
a general review on how compliance has generally been included within lumped
models.
One noteworthy feature is that the three-element windkessel model of Fig. 3D [10,
16, 21, 28, 30, 35] gives rise to a differential equation of the same form as equation
(18), except the terms have a significantly different meaning. The equation for this
circuit is:
dp
1
R1 R2 C1 dq
R 2 C1
+q =
+p ,
R1 + R2 dt
R1 + R 2
dt
74
METTE S. OLUFSEN AND ALI NADIM
10
8
q
6
4
2
0
−2
−2
−1
t
0
1
2
Figure 4. Flow as a function of time for a pressure drop p at time
t = 0. The dotted curve is for the windkessel model and the solid
curve is for the rigid vessel.
which also can be written in the form of equation (18):
dp
dq
+ q = α λp
+p .
λq
dt
dt
Integrating this equation to solve for q(t) gives:
Z t
λp
λp
α
1−
e−(t−t̃)/λq p(t̃) dt̃ + α p.
q=
λ
λ
λq
q
−∞ q
The above analysis is similar to the one carried out by Bird et al. [6] for viscoelastic
flow.
Now, consider the response of the flow to a sudden pressure drop occurring at
time t = 0:
p0 for t ≤ 0
p=
0 for t > 0.
The above equation for q for t > 0 reduces to
λp
e−t/λq .
q =α 1−
λq
(31)
The fundamental difference between the three-element windkessel model and the
rigid vessel lumped model lies in the relation between the relaxation time λ q and the
retardation time λp . For the rigid vessel model, the retardation time is significantly
smaller than the relaxation time (λp λq ), whereas for the three-element windkessel model the relaxation time is smaller than the retardation time (λ q < λp ). As
a result the response q will have the form shown in Fig. 4. The important point
to note is that for the rigid vessel the flow will always be positive (see solid line in
Fig. 4). It will start at πp0 /8, and at t = 0 it will suddenly jump to πp0 (1−λp /λq )/8,
from which it will decrease exponentially to zero. The reason for the initial jump
is that when considering a step change in pressure that forces the flow, that step
change will result in a delta-function forcing in the equation for q and hence an
impulsive change in flow. For the three-element windkessel model, where λ p > λq ,
the term in the parentheses in equation (31) becomes negative. Therefore, at t = 0
the flow will jump to a negative value and then increase exponentially to zero (dotted line in Fig. 4). The latter is a consequence of the capacitor: consider an elastic
vessel at an initially high pressure p0 where the pressure is suddenly turned off at
ON LUMPED MODELS FOR BLOOD FLOW
75
t = t0 . As a result the vessel will try to shrink and the flow could reverse. A similar
phenomenon cannot take place in a rigid vessel.
Therefore, although the derivations of the rigid vessel lumped model and the
three-element windkessel model lead to a differential equation of the same form, they
cannot be interpreted the same way. In fact, if one tries to match the coefficients
of the windkessel model to the ones from the rigid vessel model one will end up
with a negative resistance and compliance. However, the NS-based derivation of
the rigid vessel lumped model can provide information on the two resistances and
the inductance, leaving just one parameter (the capacitance) that needs to be found
from studying the properties of the arterial wall.
3.3. Compliance Models. None of the models discussed in the previous sections
included elasticity, which in the circuit analogies amounts to adding capacitors.
Several suggestions and derivations of lumped models are based on fluid flow in
elastic vessels, two such models are [5, 15]. In this section we provide a summary
of how compliance can be treated by reviewing the work discussed by Cheer and
Keener. These are, to our knowledge, the only attempts to include compliance in
these simplified models.
The model discussed by Keener and Sneyd departs from equation (1) by simply
assuming that the pressure is a function of time only. That is, the inertial and viscous terms in the momentum equation are neglected altogether, yielding a uniform
pressure within the vessel, which is assumed to be compliant. When combined with
the equation for conservation of volume this yields
∂u
dp
= 0,
(32)
c +A
dt
∂x
where it is assumed that A(p) = A0 + cp, in which c is a local compliance. Solving
for u(x, t) by integrating from x = 0 to x = L and multiplying by A0 , the crosssectional area at zero transmural pressure, gives
dp
p
q(t) = θ(p) + .
(33)
dt
R
Equation (33) corresponds to the two-element windkessel model, where the compliance of the vessel is given by
Z L
A0 c
dx,
θ(p) =
A
+ cp
0
0
the inflow into the vessel is given by q(t) = A0 u(0, t), and the resistance R acting
on the flow out of the vessel is given by A0 u(L, t) = p/R. This model can also be
obtained by adding a capacitor to equation (21) represented by the circuit shown
in Fig. 3A.
To our knowledge, a similar derivation has not been made for either the threeor the four-element windkessel models. These models can be obtained by simply
adding a capacitor to the circuit shown in Fig. 3C. The four-element windkessel
model (Fig. 3E) can be obtained directly, while the three-element model (Fig. 3D)
can be obtained by letting L1 → ∞, in which case all the flow will go through
the resistance R1 and none will go through the inductor. The equation for the
four-element windkessel model (see Fig. 3E) is given by
C 1 L1 R 1 R 2
dq
d2 p
dp
d2 q
+L1 (R1 +R2 ) +R1 R2 q = C1 L1 R2 2 +(L1 +C1 R1 R2 ) +R1 p.
2
dt
dt
dt
dt
(34)
76
METTE S. OLUFSEN AND ALI NADIM
Comparing this model with the one without the capacitance it is seen that the
added capacitance gives rise to terms involving second-order derivatives in both
p and q as well as an additional contribution to the coefficient of the first-order
derivative of p.
Letting L1 → ∞ and integrating once with respect to t yield the differential
equation for the three-element windkessel model:
dp
dq
+ (R1 + R2 )q = C1 R2
+ p.
(35)
C1 R1 R2
dt
dt
Alternatively, letting R1 → ∞ in equation (34) gives a model similar to the one
shown in Fig. 3B with an added capacitor. This model can be written as
d2 q
dq
dp
+ L1
+ R 2 q = C 1 R2
+ p.
(36)
2
dt
dt
dt
Finally, a last approach to add compliance is the one suggested by Berger in [5]. In
this paper, the point of departure is the axisymmetric flow in a vessel as described in
equation (1) combined with equation (32) accounting for conservation of volume.
In the derivation by Berger, viscosity and inertance are both included. In the
equation for conservation of volume leakage through the vessel is also allowed,
which is actually essential to the derivation. This leakage should be interpreted
as the amount of blood lost due to vessels branching off from the vessel we study.
Integrating the Womersley equation (1) over the cross-sectional area, assuming a
Poiseuille velocity profile, and adding leakage to equation (32) for conservation of
volume give:
ρ ∂q
8µ
∂p
=
+
q
−
∂x
A ∂t
πR4
∂q
∂A ∂p
−
=
+ w0 p
∂x
∂p ∂t
where ∂A/∂p is the compliance and w 0 is the leakage per unit length which is
equivalent to conductance. Instead of computing the lumped equation directly from
the circuit, the analogy to signal transmission through a uniform cable is used. In
this analogy pressure is equivalent to voltage, flow to current, ρ/A to inductance
per unit length, 8µ/πR4 to resistance. With the above interpretation the equations
above can be represented by the circuit shown in Fig. 5. The equations for this
circuit are given by:
dq
p − p1 = R 1 q + L1
dt
dp1
C1
= q − q̃
dt
1
q̃.
p1 − 0 =
G1
Eliminating p1 and q̃ gives
C 1 L1 R 2
d2 q
dq
dp
+ G1 p = C1 L1 2 + (C1 R1 + L1 G1 ) + (1 + R1 G1 )q.
dt
dt
dt
Note that G1 can be interpreted as a conductance for the leakage current. Its
inverse plays the same role as the resistance R2 would in Fig. 3D or 3E. If G1 is
set to zero (no leakage through the wall) or the corresponding resistance R 2 goes
to infinity, the model fails because the flow path would end in the capacitor and as
a result there will be no net flow through the system. In other words, it might be
C1
ON LUMPED MODELS FOR BLOOD FLOW
R1
L1
p
77
p1
q
q − q̃
C1
G1 q̃
Figure 5. Circuit obtained from the model by Berger. The
lumped model represented by this circuit is based on the fluid flow
in an elastic vessel, with leakage through the wall.
better to interpret 1/G1 as the peripheral resistance and always keep it in Berger’s
model. If G1 is replaced with 1/R2 in the last equation, we arrive at a circuit which
is second order in flow q and first order in pressure p. This is similar to equation
(36) although the latter only has one resistance rather than two.
In summary we conclude that if compliance is to be included in the fluid dynamic
derivation, there is to our knowledge no simple way to obtain a lumped model
without making a number of assumptions. One must assume that the fluid is
inviscid and that the inertia is negligible, or simply add a capacitor to one of the
models derived for flow in a rigid vessel, or allow for leakage through the vessel
wall. In any case, based on our analysis of the time-scales lumped models are
mainly useful for studying flow in the smaller vessels with a radius less than 0.5
cm.
4. Discussion. In this paper we have derived a number of lumped models based on
the axisymmetric time-dependent fluid flow equation for a rigid vessel also known
as Womersley’s equation. Assuming that the vessel is rigid and the flow is laminar,
we were able to make a series expansion of the solution of Womersley’s equation
to obtain a number of lumped models. Based on studying the characteristic timescales for this problem we can conclude that for arteries with a radius smaller
than 0.2 cm it is possible to assume that pressure is proportional to flow. The
circuit representing such a vessel would simply contain a resistance and no other
elements. In other words, effects due to inertia can completely be ignored. Since
such small arteries are typically rigid (although there are exceptions) but do provide
resistance, it will be unnecessary to add a capacitor to the model to account for
elasticity. However, as shown by Keener and Sneyd [15] it is possible to incorporate
elasticity using the two-element windkessel model; that is, by adding a capacitor to
the circuit shown in Fig. 3A. It should be noted that the derivation by Keener and
Sneyd is somewhat artificial, it includes neither viscosity nor inertia.
For vessels with a radius between 0.2 and 0.5 cm, it is necessary to add additional
elements to the model. The minimum model that would take some of the effect
of inertance into account is the first-order model (23) that includes a resistor and
inductor in series (see Fig. 3B). As for the very small vessels, this model can be
obtained either from including an additional term in the residue method for the
inversion integral, but neglecting the small retardation time.
For vessels with a radius larger than 0.5 cm, we conclude that a higher-order
model should be used. By higher-order, we mean a model that includes contributions from more of the residues. If two terms are included, one can still ignore
78
METTE S. OLUFSEN AND ALI NADIM
the second-order derivative in pressure, yielding a model with two derivatives (i.e.
second-order) in flow and one derivative (i.e. first order) in pressure. A circuit that
can represent this model would be equivalent to the fourth-order windkessel model,
but without the capacitor.
Finally, for vessels with a radius larger than 1.5 cm, more terms in the expansion
are needed to get an accurate lumped model of the flow. Therefore, we conclude that
for large vessels (e.g., the aorta) it would be more appropriate to use distributed
models (one-, two-, or three-dimensional models) for simulating blood flow and
pressure. This fits well with our intuitive understanding that in such vessels the
fluid dynamics is more complex and spatially varying. In that case, one option is
to use the windkessel model as a boundary condition at the outflow for the large
vessels, as discussed for instance in the papers [19, 27].
Our derivation provides some justification for the recent observation that the
four-element windkessel model (including inductance) suggested by [29] fits the
data better than the three-element windkessel model. It should be noted that
lumped models are frequently used to describe the cardiovascular system as a whole,
including all vessels from the aorta to the arterioles. In our derivations, however, we
considered flow in a single vessel and its corresponding lumped description. In our
view, a more accurate but obviously more complex model of the systemic arteries
can be constructed by considering the bifurcating network of branched arteries and
describing each element in that network by its appropriate lumped model. Since
coupling such low-order models will quickly result in a high-order model, it may
be justified to simplify the description and use a low-order model for the network
as a whole. It is evident though that by coupling individual elements that possess
resistance, compliance and inductance, the whole network should also possess such
elements in its overall lumped description.
Acknowledgements. Mette Olufsen was supported by a modular grant from the
National Institute of Health #R03-AG20833-01 and by a Faculty Research and
Development Fund Grant from North Carolina State University SPS #0064-8604.
Appendix A. Residues for M (t). The residues of (K(s)/s)est are computed
herein. To do so it is advantageous to write
√
√
√ π i sJ0 (i s) − 2J1 (i s)
K(s) st
√
√
e = 2
est .
(37)
s
s
i sJ0 (i s)
From the above expression, it can be seen that√(K(s)/s) est has a simple pole at
s = 0 and an infinite number of poles where J0 (i s) = 0. The residue at the simple
pole at s = 0 can be found to be π/8 by series expansion for small values of s:
√
π
2J1 (i s)
√
K(s) =
1− √
s
i sJ0 (i s)
√
√
√
(i s)4
11(i s)6
π
(i s)2
+
+
+ ...
=
1− 1+
s
8
48
3072
2
3
π s
s
11 s
=
+
−
+ O(s4 )
s 8 48
3072
π πs 11πs2
−
+
+ O(s3 ).
=
8
48
3072
ON LUMPED MODELS FOR BLOOD FLOW
79
√
The remaining residues (where J0 (i s) = 0) can be found as follows: denote the
zeros of J0 (β) by βn , n = 1, 2, 3, . . ., then the poles for (K(s)/s) est are at 0 and at
points where
√
βn = i s n
⇔
sn = −βn2 .
Starting with equation (37) and evaluating the residue at sn = −βn2 yields:
"
#
2
−2J1 (βn )
π
e−βn t
Res(sn ) =
√ d
5
βn ds J0 (i s) s=−β 2
n
2
π
−2J1 (βn )
=
e−βn t
5
βn J1 (βn )/(2βn )
2
=
−4πe−βn t
.
βn4
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Received on Jan. 9, 2004. Revised on Feb. 18, 2004.
E-mail address: [email protected]
E-mail address: [email protected]